Stronger Communities through Better Transit Act This bill requires the Department of Transportation (DOT) to establish a grant program to support operating projects for public transportation and related service improvements, particularly in underserved communities and areas of persistent poverty. Specifically, the bill requires DOT to allocate funding under the program for urbanized areas, states, and Indian tribes that are recipients of funds under either the Federal Transit Administration's (FTA's) Urbanized Area Formula Funding program or Formula Grants for Rural Areas program. Eligible recipients may use funding for operating costs associated with projects that improve public transportation service for transit-dependent populations and support increased transit ridership (e.g., service expansion, information technology enhancements, and workforce development). The bill requires DOT to apportion the funding so that recipients receive funds that are proportional to their share of operating costs. The bill also provides for an increased federal cost share for operating assistance for projects or programs carried out in areas of persistent poverty or underserved communities. DOT must set up a multimodal access measurement interface for public agencies to aid transit agencies in determining and reporting on access to jobs and essential services. A grant recipient must also (1) report specific information to the FTA for inclusion in the National Transit Database, and (2) survey transit riders and non-riding residents regarding transit service improvements. Further, the bill expands the purposes of the public transportation programs to include supporting public transportation's role in combating climate change through growing/retaining transit ridership.